What is the rule for multiplying exponents with the same base?

When multiplying exponents that share the same base, the rule is to add the exponents together. This arises from the definition of exponents themselves. For example, if you have a base ( a ) and you are multiplying ( a^m ) by ( a^n ), this is essentially taking ( a ) to the power of ( m ) and multiplying it by ( a ) to the power of ( n ):

[

a^m \times a^n = a^{m+n}

]

This rule allows us to keep the base consistent while effectively combining the powers being applied to it. The logic behind this is that when you multiply terms with the same base, you are effectively increasing the total number of times the base is being multiplied by itself, hence leading to the addition of the exponents.

For instance, if ( m = 2 ) and ( n = 3 ), multiplying the bases gives you:

[

a^2 \times a^3 = a^{2+3} = a^5

]

Thus, the correct understanding is that you add the exponents when multiplying two exponents with the same base, confirming the answer as adding them is the established